Dirac structure

Definition

A Dirac structure on a Riemannian manifold $(M,g)$ is a quadruple $(E,c,\nabla ^{E},\nabla ^{M})$ where:

• $E$ is a Hermitian vector bundle on $M$
• $c:T^{*}(M)\to End(E)$ is a self-adjoint Clifford structure on $E$
• $\nabla ^{M}$ is a metric connection on $M$
• $\nabla ^{E}$ is a Hermitian connection on $E$ compatible with Clifford multiplication

A geometric Dirac structure is a Dirac structure where the metric connection $\nabla ^{M}$ is the Levi-Civita connection.